#7677946

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#45412	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,14)(12,16)(13,15)(17,18,23,20)(19,27,29,24)(21,25,31,30)(22,28,32,26);(1,3,5,8)(2,6,4,7)(9,11,13,16)(10,14,15,12)(17,19,23,29)(18,24,20,27)(21,26,31,28)(22,25,32,30);(9,12)(10,11)(13,14)(15,16)(17,18,23,20)(19,27,29,24)(21,30,31,25)(22,26,32,28);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,20,23,18)(19,27,29,24)(21,30,31,25)(22,28,32,26);(1,2,5,4)(3,6,8,7)(17,21)(18,25)(19,28)(20,30)(22,24)(23,31)(26,29)(27,32);(1,4,5,2)(3,7,8,6)(17,22,23,32)(18,26,20,28)(19,25,29,30)(21,24,31,27)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585142);